(I would describe myself as an illiterate physics enthusiast, so I hope you'll forgive me if my ignorance is borderline offensive.)
If I've understood anything of the concept of time dilation, your perception of time slows down as you're approaching the speed of light.
And I was wondering if you could revert the process and experience time actually going faster if you went out of your way to not move. Suppose you're standing atop of the Sun, observing the Earth revolving, –and for the purpose of this thought experiment, you have no mass because I've come to understand gravity also bends time– am I right to think you'll find earthlings living ever so slightly slower? Then, repeat the process and place yourself atop of the center of the galaxy, looking again at Earth revolving around a revolving star. Even slower?
And then… I realized at this point there lacks an higher traditional referential we could use to "not move" even more as to see Earth's years pass as seconds. Considering time (rather your perception of it) stops when moving at lightspeed, is there an exact opposite phenomenon occurring when at "absolute zero speed" that would make time seem to go infinitely fast? And how would that exist considering any speed is measured relative to something else?
Best Answer
In special relativity there is no way you can see someone elses time going faster. This is because in SR all motion is relative. There is no notion of an absolute state of rest. In dmckee's example of the muon experiment, we see time moving more slowly for the muons. However the muons (if they were sentient) would see time moving more slowly for us. This apparent paradox is the cause of much (wasted!) time discussing the twin paradox.
However, in General Relativity things are different. The deeper into a gravitation field you go the more slowly your time runs compared to an observer well away from the field. So if you jumped into a powerful rocket and hovered just above the event horizon of a black hole you would indeed see time moving faster for your friends who stayed well away from the black hole. The ratio between your time, $\tau$ and the external observers time, $t$, is given by (for a stationary black hole):
$$ \frac{\tau}{t} = \sqrt{ 1 - \frac{r_s}{r}} $$
where $r_s$ is the radius of the event horizon and $r$ is your distance away from the centre of the black hole. If you go right to the event horizon, i.e. $r = r_s$ the the ratio falls to zero. That means the external observers see your time stop and you see the external observer's time going infinitely fast.